The WYD scalar-curvature Schur-increasing conjecture near the BKM metric

Let $p\in(1,+\infty)$, and let $\operatorname{Scal}_{\mathrm{WYD}(p)}$ denote the scalar curvature of the Wigner–Yanase–Dyson metric with parameter $p$. Let $\succ$ denote majorization and let $I=(1,1+\varepsilon)$ for some $\varepsilon>0$.

WYD scalar-curvature conjecture. There exists $\varepsilon>0$ such that, for every $p\in I$, the scalar curvature of the WYD$(p)$ metric is a Schur-increasing function.

The conjecture proposes a family of monotone metrics with Schur-increasing scalar curvature near the BKM endpoint $p=1$. The paper notes that, apart from the constant-curvature WY metric and the non-monotone SLD example, no such examples were known; the assertion is left open.